Thinking out of the box, I think the problem can be solved with logistics and expansion.

Why not send two airplanes up, at the same runway, but opposite way? One airplane at each side of the runway. Airplane 1 starts to take off, when just leaving runway, airplane 2 starts right away. So on it goes. You have to direct the traffic in different way, since airplanes coming out and in both ends, but you have doubled the capacity. (also agree to route planes to nearby airports).

Other way is to build a landing stip on top of the existing one. 500 meters in or so, you can start building the other landing stip, on top.

I leave the details to the experts, but think this might work. Very unconventional solution, but sometimes it needs to be like that.

Car electricity is a body-board vehicle electronic control devices. The car alternator can convert mechanical energy to electrical energy in the form of alternating current. The car starter and voltage regulator perform the similar function for the vehicle. The pressure sensor used to transfer electrical signal. Installing the new head light for good illumination is a good choice!

London already has 5 airports. It should probably use its existing airport capacity fully before expanding or building more. The worst idea is to build a new airport in the Thames estuary - that is completely unnecessary, and would have extremely high monetary and environmental costs.
For example, London Stansted has a 10000 foot runway but no transatlantic flights. Why can't they route some planes there instead of building new ones? That would save money in this austerity time anyway.
Perhaps they should invest in direct fast rail links between the airports so that passengers can easily connect. Maglev is hugely expensive - perhaps a regular express train would do the trick.
Just my opinion as an outsider.

You don't necessarily need to expand it geographically (to the north or the west). You can expand the airport by moving the runways "up" vertically and taking advantage of vertical opportunities by relocating much of what is competing for space (including the terminals, transit, roadways, etc) to "under" the new runway levels.

Another thought ... Heathrow's unique capacity constraints and means it captures an unusually high share of especially noisy types of aircraft like the 747, which was until the A380 the best aircraft to take maximum advantage of its expensive slots (and hence British Airways is the biggest 747 operator in the world). Other airports like LaGuardia in New York also have flight paths across huge numbers of homes but capture far fewer complaints. It is possible that greater runway capacity could allow airports to ease up on the pressure to use the biggest plane possible, which would allow a better mix of quieter planes. Of course, as 747s are retired in favor of new-generation aircraft, the noise will reduce on its own anyway.

Most Heathrow based complaints come from NIMBY's. After that, this letter column is filled with letters which decided to use this article as a dare to come up with better ideas than the Economist, ones own ideas are always better right?
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One of the few good letters in there is the one which calls this idea a no-brainer, which frankly it is. Now all we need is the political courage to do it. Perhaps if the major parties decided to do this together, Olympics style then no one could be voted out by NIMBYs.

I think the best suggestion so far is a maglev between gatwick and heathrow. They can easily operate as one airport.

THis letter is off the mark, though - "If you want an example of a country that has got this right, simply look to Germany. It uses Düsseldorf, Frankfurt and Munich as major hubs with near-enough equal capacity. Is it a coincidence that its economy has been one of the most resilient in Europe?"

Firstly, Germany is huge compared to the UK. Secondly, Germany's economy is much better spread over the country relative to the UK where more than half of the GDP is generated from the midlands downward, all within 1 1/2 hours by rail from london.

The idea to re-align the runways is a non-starter -- Heathrow once had 3 runways going in different directions, but those were scrapped because as aviation matured, people realized that in many areas the winds typically blow in one direction, and it is by far safest to take off and land aircraft either with or into the wind. Building a new runway askew to prevailing winds in an area is simply a non-starter, and would make that runway far less useful since on gusty days it might not be usable.

That said it may be possible that efforts could be made to have more aircraft movements take place coming in from and departing out to Heathrow's west. When a new runway was added at Logan Airport in Boston, a tall hotel was intentionally built right at the end of the runway to ensure that it would never be possible to route planes over downtown Boston to get to that runway, but rather than they would have to come in from and go out over the water.

Actually, Tokyo's airports have a conundrum: Their two most strongest prevailing winds are Northwesterly and Southwesterly, ie. almost perpendicular to each other.
In Narita Airport, left-wing opposition have blocked the site for the planned runway B that was supposed to cater for the Northwesterly wind.
Today, Haneda has become a better Airport, with 2 runways each for Northwesterly (A and C) and Northeasterly (B and D) winds.

In china they would have had it build by now. Instead here we have over paid benefit claiments in paraliment wininging on about some other lower paid benefits claiments while agreeing to do nothing for the length of parlaiment for fear a passing plane may pop the bubble that is there property price a product of some state sponsered ponze scheme -)